Armature's Not Fully Connected to Mesh?
The problem is that the 'Spine' is  also affecting the head and the arms. If you remove the arms and head from this vertex group, it works just fine.

In weight paint mode choose the 'Spine' vertex group. Use the subtract brush to remove the weights from the arms and head. Then test with different poses and adjust the weights where needed.

Here is a screenshot of how it should look like (although some more tweaking is needed :) ):